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Psychometrika

, Volume 56, Issue 1, pp 69–75 | Cite as

A simple multivariate probabilistic model for preferential and triadic choices

  • Kenneth Mullen
  • Daniel M. Ennis
Article

Abstract

Multidimensional probabilistic models of behavior following similarity and choice judgements have proven to be useful in representing multidimensional percepts in Euclidean and non-Euclidean spaces. With few exceptions, these models are generally computationally intense because they often require numerical work with multiple integrals. This paper focuses attention on a particularly general triad and preferential choice model previously requiring the numerical evaluation of a 2n-fold integral, wheren is the number of elements in the vectors representing the psychological magnitudes. Transforming this model to an indefinite quadratic form leads to a single integral. The significance of this form to multidimensional scaling and computational efficiency is discussed.

Key words

multivariate triads duo-trio canonical quadratic forms choice models multidimensional scaling preference unfolding 

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References

  1. Ashby, F. G., & Perrin, N. (1988). Toward a unified theory of similarity and recognition.Psychological Review, 95, 124–150.Google Scholar
  2. Coombs, C. H. (1964).A theory of data. New York: Wiley.Google Scholar
  3. David, H. A., & Trivedi, M. C. (1962).Pair, triangle and duo-trio tests (Technical Report 55). Blacksburg, VA: Virginia Polytechnic Institute, Department of Statistics.Google Scholar
  4. Dennis, J. E., Jr., & Schnabel, R. B. (1983).Numerical methods for unconstrained optimization and nonlinear equations. New Jersey: Prentice Hall.Google Scholar
  5. De Soete, G., Carroll, J. D., & DeSarbo, W. S. (1986). The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data.Journal of Mathematical Psychology, 30, 28–41.Google Scholar
  6. Ennis, D. M. (1988b). Toward a universal law of generalization.Science, 242, 944.Google Scholar
  7. Ennis, D. M. (1988b). Confusable and discriminable stimuli: Comment on Nosofsky (1986) and Shepard (1986).Journal of Experimental Psychology, 4, 408–411.Google Scholar
  8. Ennis, D. M., & Mullen, K. (1986a). A multivariate model for discrimination methods.Journal of Mathematical Psychology, 30, 206–219.Google Scholar
  9. Ennis, D. M., & Mullen, K. (1986b). Theoretical aspects of sensory discrimination.Chemical Senses, 11, 513–522.Google Scholar
  10. Ennis, D. M., Mullen K., & Frijters, J. E. R. (1988). Variants of the method of triads: Unidimensional Thurstonian models.British Journal of Mathematical and Statistical Psychology, 41, 25–36.Google Scholar
  11. Ennis, D. M., Palen, J. J., & Mullen, K. (1988). A multidimensional stochastic theory of similarity.Journal of Mathematical Psychology, 32, 449–465.Google Scholar
  12. Ennis, D. M., Mullen, K., de Doncker, E., & Kapenga, J. (1989). A general mathematical model for triadic and preference choices. Unpublished manuscript. Philip Morris Research Center, Richmond, Virginia.Google Scholar
  13. Frijters, J. E. R. (1979). Variations of the triangular method and the relationship of its unidimensional probabilistic models to three-alternative forced choice signal detection theory models.British Journal of Mathematical and Statistical Psychology, 32, 229–241.Google Scholar
  14. Genz, A. C., & Malik, A. A. (1980). Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region.Journal of computing and Applied Mathematics, 6, 295–302.Google Scholar
  15. Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables.Biometrika, 48, 419–426.Google Scholar
  16. Johnson, N. L., & Kotz, S. (1970).Continuous univariate distributions — 2, New York: Wiley.Google Scholar
  17. MacKay, D. B. (1989). Probabilistic multidimensional scaling: An anisotropic model for distance judgments.Journal of Mathematical Psychology, 33, 187–205.Google Scholar
  18. Mullen, K., & Ennis, D. M. (1987). Mathematical formulation of multivariate Euclidean models for discrimination methods.Psychometrika, 52, 235–249.Google Scholar
  19. Mullen, K., Ennis, D. M., de Doncker, E., & Kapenga, J. A. (1988). Models for the duo-trio and triangular methods.Biometrics, 44, 1169–1175.Google Scholar
  20. Nosofsky, R. M. (1986). Attention, similarity and the identification-categorization relationship.Journal of Experimental Psychology: General, 115, 39–57.Google Scholar
  21. Nosofsky, R. M. (1988). On exemplar-based representations: Comment on Ennis (1988).Journal of Experimental Psychology: General, 117, 412–414.Google Scholar
  22. Ruben, H. (1960). Probability content of regions under spherical normal distributions, I.Annals of Mathematical Statistics, 31, 598–619.Google Scholar
  23. Ruben, H. (1962). Probability content of regions under spherical normal distributions, IV.Annals of Mathematical Statistics, 33, 542–570.Google Scholar
  24. Ruben, H. (1963). A new result on the distribution of quadratic forms.Annals of Mathematical Statistics, 34, 1582–1584.Google Scholar
  25. Shepard, R. N. (1986). Discrimination and generalization in identification and classification: Comment on Nosofsky.Journal of Experimental Psychology: General, 115, 58–61.Google Scholar
  26. Shepard, R. N. (1987). Toward a universal law of generalization for psychological science.Science, 237, 1317–1323.Google Scholar
  27. Shepard, R. N. (1988a). Time and distance in generalization and discrimination: Comment on Ennis (1988).Journal of Experimental Psychology, General, 117, 415–416.Google Scholar
  28. Shepard, R. N. (1988b). Toward a universal law of generalization: Response.Science, 242, 944.Google Scholar
  29. Thurstone, L. L. (1927). A law of comparative judgement.Psychological Review, 34, 273–286.Google Scholar
  30. Torgerson, W. S. (1958).Theory and methods of scaling. New York: Wiley.Google Scholar
  31. Zinnes, J. L., & MacKay, D. B. (1987). Probabilistic multidimensional analysis of preference ratio judgements.Communications and Cognition, 20, 17–44.Google Scholar

Copyright information

© The Psychometric Society 1991

Authors and Affiliations

  • Kenneth Mullen
    • 1
  • Daniel M. Ennis
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of GuelphCanada
  2. 2.Philip Morris Research CenterRichmond

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